Limitations in the Backanalysis of Shear Strength From Failures

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Limitations in the Back-Analysis of Strength from Failures

Rick Deschamps1 and Greg Yankey2,

Nicholson Construction Company, 12 McClane Road, Cuddy, PA

FMSM Engineers, 1409 North Forbes Road, Lexington KY

ABSTRACT

Stability failures are often "back analyzed" in an attempt to estimate the operative shear strength.

In fact, back-analysis is commonly believed to be one of the most reliable ways to estimate soil

and/or rock strength. However, this paper provides examples to illustrate specific situations in

which back-analysis of failures can lead to misinterpretation of strength. Examples from earth

and concrete gravity dams are used, and consideration is given to both 2-D and 3-D idealizations.

The cases demonstrate that interpreted strength can be in significant error, and in practically all

cases the errors are unconservative. Finally, this paper illustrates that back-analysis is reliable

only when the model and all assumptions are reasonable and accurate representations of the real

system.

INTRODUCTION

Back-analysis is an approach commonly used in geotechnical engineering to estimate operable

material parameters in-situ. This approach is popular because there are significant limitations in

the use of laboratory and in-situ test results to accurately characterize a soil profile. Numerous

studies have demonstrated the use of back-analysis techniques for the determination of soil

parameters. There are also several publications that describe limitations of back analyses

(Leroueil and Tavenas, 1981; Azzouz et.al., 1981; Leonards, 1982; Duncan and Stark, 1992;

Gilbert et. al. 1998; Tang, et. al. 1998; Stark et. al. 1998). Although these publications describe

many of the pitfalls of back-analyses, it is the authors’ experience that these facts are not fully

appreciated by many practicing engineers. Accordingly, the objective of this paper is to illustrate

how different assumptions made in the back-analysis of failures can influence the interpretedshear strength. A recognized fact that must be recognized is that a conservative design

assumption is unconservative when used in back-analysis because other values of shear strength

may be conservatively in design causing the back calculated strength to be over estimated. As

engineers, we commonly build some conservatism into the selection of design parameters. This

tendency, often subconscious, leads to unconservative interpretations of strength in using back-

analysis. Moreover, the models that are typically employed in geotechnical engineering embody

conservative assumptions. Accordingly, the models can also lead to unconservative results when

used in back-analyses, and therefore, must be factored into interpretations. What is conservative

in design is unconservative in back-analysis, and vice versa. In this paper, some of the factors

that complicate the use of back-analysis are described first. Subsequently, specific project

examples are used to illustrate the magnitudes of potential errors.

FACTORS THAT INFLUENCE INTERPRETED SHEAR STRENGTH DURING BACK-

ANALYSIS.

1. The relative strength of materials in heterogeneous profiles impacts interpretations of the

target material strength. Often it is desirable to back-calculate the strength of a weak

layer or seam. However, to accurately back-calculate the strength of the desired material,

the strength of all other materials must be known.



2. The slip surface analyzed must be the same as the actual rupture surface to effectively

back-calculate the strength of a deposit. Leonards (1982) describes several cases where a

back-analysis was reported to show a safety factor near 1.0 using laboratory or in-situ

determined shear strengths, but the failure surface in the analysis is not consistent with

the actual rupture surface. Conclusions are sometimes drawn that the limit equilibrium

method used with shear strengths obtained by some specific approach can predict the

failure, but not the surface (e.g. Skempton 1945). This logic is flawed because the actualslip surface will demonstrate different back-calculated strengths than the "critical"

surface obtained from analyses

3. . Furthermore, a significant challenge is that the actual rupture surface may be known at

only a few locations, if any.

Another example of uncertainty in the slip surface is the presence of a tension crack, itsdepth, and whether it is full of water. For materials that are characterized as having

relatively high cohesive strength, the assumption of a tension crack will have a significant

influence on interpreted stability, and therefore, the back-calculated strength.

4. Knowledge of the pore water pressure is required to determine effective stresses, andtherefore strength. Sometimes there is pre-failure piezometric data at select locations,

sometimes measurements are made post failure, and other times pore pressures are

estimated. However, it must be recognized that the actual distribution of water pressure

can be complicated, and that the operable pore pressures at failure, including shear-

induced pore pressures, cannot be reliably measured.

5. Practically all slopes have a three dimensional component. Neglecting this component in

back-analysis will lead to an overestimation of strength. However, “end effects” are not

easily accounted for because the influence on stability can vary over a broad range.

Azzouz et. al, (1981); and Stark and Eid, (1998) provide estimates of the influence of

three-dimensional factors in stability analyses. In general, for soil slopes and

embankments the end effects appear to increase stability by from 5 to 30 percent.

6. Progressive failure in strain softening materials will also affect interpretation of strength.

If the back-calculated strength is to be used for similar slopes in similar stratigraphy, the

back-calculated operable strength may be useful (Duncan and Stark, 1992). However, if

the loading condition induces a significantly different stress path, the back-calculated

strength from another stress path or geometry may be misleading. For example, using

strength back-calculated from slope failures may not be appropriate in foundation design.

7. Mohr-Coulomb strength is defined by a friction angle and a cohesive intercept.

Determination of these parameters individually is typically not possible unless significant

redundant data is available (Duncan and Stark, 1992).

PROBLEM ILLUSTRATED

Relative safety is typically quantified by use of a safety factor that is defined as some ratio of

resisting forces (and/or moments), to the forces required for equilibrium (driving forces). This

relationship can be idealized for the case of general wedge surface for a slope using Eqn 1,

wherein the numerator is related to the strength and the denominator to the mobilized stresses.



( ){ }

∑∑ −+

=

ii

iiiii uc

SF

l

l

τ

φ σ tan (1)

Where:

c = cohesive intercept

σ = normal stress

u = water pressure

φ = friction angle

τ = mobilized shear stress

l = length of rupture surface in layer

i = wedge being considered

Consider Eqn 1, if the strength parameters (c and φ) are underestimated, the safety factor is

reduced, and therefore, the analysis is conservative. Likewise, if the water pressure is

overestimated, the computed safety factor is also conservative. Furthermore, if there is resistance

from three-dimensional effects that is not accounted for in the numerator of Eqn 1, the safetyfactor is underestimated, so the results are conservative. As engineers, we often take solace in

these "hidden" conservatisms such that we tend to lean to the conservative side of data containing

scatter.

In the back-analysis of a failure, the assumption is made that the safety factor is 1.0 so that

resisting forces equal the driving forces. For the simple case of a two-wedge system with no

cohesion, Eqn 1 becomes:

{ } { } 221122221111 tan)(tan)( llll τ τ φ σ φ σ +=−+− uu (2)

The right side of Eqn 2 is dictated by equilibrium and can be considered "known" for a given

geometry and rupture surface. The objective of back-analysis is to determine the strength

components on the left side of Eqn 2. Equating the resisting forces with the driving forces, by

fixing the safety factor at 1.0, leads to the condition that conservative design assumptions are

unconservative in back-analysis, as is illustrated by the following scenarios.

• Assume that it is desirable to back-calculate the strength of a weak material (Layer 2) from a

failure in which there is high confidence in both the failure surface location and the water

pressure. Given that the right side of Eqn 2, and σ1 and σ2 are obtained from equilibrium, an

estimate of φ1 is required in order to calculate φ2. Note that because of the imposed equality,

the magnitude of φ2 must increase as the magnitude of φ1 decreases. Therefore, if φ1 is

underestimated (typically a conservative assumption), φ2 will be overestimated (an

unconservative result).

• Similarly, overestimating the water pressure (typically conservative) will reduce the normal

effective stress and lead to a larger (unconservative) back-calculated shear strength in order to

satisfy Eqn 2.

• Virtually all slope failures possess a three dimensional aspect that is commonly neglected.

To appropriately describe this condition, another resistance term should be included on the

left side of Eqn 2. However, neglecting the three-dimensional resistance (normally a



conservative assumption) leads to higher back-calculated strengths (unconservative result) in

order to satisfy the equality.

These are just a few simple examples that illustrate a general point: assumptions that are

conservative in design are unconservative in back analysis. Importantly, as engineers we are

generally conservative by nature in both our models and in our parameters selection.

Accordingly, we must resist this tendency or risk unconservative consequences when performing back-analysis.

QUANTITATIVE EXAMPLES OF POTENTIAL ERRORS IN BACK-ANALYSIS

Project examples are provided to illustrate the challenge and potential errors that can be present in

back-analysis for Items 1 through 4 described in the previous section. Succinct examples could

not be developed for Items 5 and 6, however, their impact is generally present. The examples are

drawn from three case histories. One of the case histories is of Grandview Lake Dam located in

Bartholomew County, central Indiana. The dam is a privately owned homogeneous earth dam

that was built to form a recreational lake and waterfront homesites. This case history is used to

illustrate the dependence that the back-calculated strength along a weak seam has on assumptions

of strength in other zones. The second case history is a stability assessment/design of a very large

(2.0 million m3) spoil pile adjacent to the Kanawha River at Marmet Lock and Dam, West

Virginia. This project demonstrates the importance of characterizing the actual rupture surface.

The third case history is Lock and Dam 10 on the Kentucky River. This is a relatively small

concrete gravity dam owned by the Commonwealth of Kentucky and built circa 1905. This

project demonstrates the importance of understanding the pore pressure distribution (uplift

pressures) along a potential slip surface and three-dimensional effects when back-calculating

strengths.

Material Strengths. Thirty years after construction Grandview Lake Dam began sliding

downstream at rates of approximately one centimeter per day along a distinct thin, planer weak

seam in the claystone foundation (See Deschamps et. al. 1999, for a more complete description of

the project). Emergency berms were placed to slow movements and back-analysis was viewed

as an important tool for estimating the strength along the planer slip surface. A cross section of

the dam at the time of failure, the location of the assumed rupture surface developed from

inclinometer measurements, and the location of the piezometric surface obtained from

piezometers is shown in Figure 1. The dam was constructed primarily of glacial till and residual

soils weathered from claystones. The first challenge was to select the operable strength of the

dam materials. There was no distinct zonation of materials in the dam, and therefore, no basis forsubdividing the dam into discrete materials. Figure 2 illustrates the results of available

consolidated undrained (CU) triaxial tests. These tests included compacted specimens (dashed

lines) and undisturbed samples obtained from Shelby tubes. Most tests were completed in 1986

(Labs A and B), while the tests performed in 1997 (Lab C) were conducted on the compacted

residual mudstone. Given the heterogeneous nature of the materials in the dam it was viewed asunlikely that additional tests would further clarify the operable strength without knowledge of

material quantities and zonation. Moreover, there was little time for detailed exploration and

testing given the emergency nature of the project.



Compacted Glacial Till and

Weathered Claystone

Claystone BedrockRupture Surface

from Inclinometer

Estimated Rupture

Surface

0 ft. 30 ft. 50 ft. 100 ft.

Figure 1. Cross-Section through Grandview Lake Dam with Piezometric Surface, Inclinometer

Locations and Assumed Rupture Surface.

Five different characterizations of embankment strength are considered here for back-analysis

purposes to characterize the weak seam strength. A lower bound, average, and two upper-boundinterpretations are shown in Figure 2.. Two upper bound strengths are used here because one

represents a higher cohesive intercept while the other a higher friction angle. Given these four

characterizations of dam strength and the conditions of Figure 1, a summary of the back-

calculated friction angles in the weak seam is shown in Table 1. The strength along the weak

seam was characterized as having a zero cohesive intercept because it was rationalized that thismaterial was at or near its residual strength because the deformations along the very thin seam

were significant, at least several inches. Moreover, there is little tendency for volume change at

residual conditions such that shear induced pore pressure changes were considered negligible.

0

50

100

150

200

0 50 100 150 200 250 300 350

Effective Confining Stress (kPa)

S h e a r S t r e s s ( k P a )

TESTING FIRM A

TESTING FIRM B

TESTING FIRM C

AVERAGE

Solid lines - Shelby tube samples

Dashed lines - Laboratory compacted samples

Upper Bound (1)

Upper Bound (2)

Lower Bound

Figure 2. Consolidated Undrained Triaxial Test Data for Grandview Lake Dam



Table 1 illustrates the range in back-calculated friction angles is from 11 to 24 degrees,with the average strength providing 18 degrees. Although this range can be viewed asextreme, it is apparent that even if a narrower range of strengths were used tocharacterize the dam, the back calculated strength would still vary over an appreciablerange. A conservative (low) estimate of embankment strength leads to a relatively highinterpretation of strength along the weak seam. Note also the significant difference inback-calculated strengths for the two upper bound cases, wherein the case with

primarily cohesive strength leads to a much lower back-calculated friction angle for thegeometry considered because of the higher shear strength of the compacted materials.

Rupture Surface. Leonards (1982) discusses the importance of accuratecharacterization of the slip surfaces in back-analysis. The example that is used here istaken from analyses completed recently for a staged construction stability assessment ofa spoil pile on a foundation containing loose sands and soft clays adjacent to a guidewallat Marmet Lock. The criterion for end of construction safety factor was 1.1. The end-of-construction design strength of the spoil material was specified as 77 kPa (1600 psf ).Based on the results of limit equilibrium analyses using circular failure surfaces, thisstrength was believed to be adequate to meet the stability criterion for failure within thespoil material. However, during numerical modeling using the program FLAC developed

by ITASCA (2002) to assess stability within the foundation, an interesting failure surfacedeveloped. Figure 3 shows a simplified cross-section of the system analyzed and thematerial properties used. The simulation included modeling the placement of theembankment in one foot "lifts," with failure occurring near the point of reaching fullembankment height The apparent critical circular surface obtained from limit equilibriumusing the program PC-STABL is shown on this figure and has a safety factor of 1.12.

Also shown on the figure with shear strain contours is the failure surface obtained fromnumerical modeling, wherein the base of the rupture surface is located within theembankment and then is redirected to the foundation immediately below theembankment when approaching the toe of the slope. In hindsight, this seems logicalbecause the lower vertical stresses near the toe leads to lower shear strengths in thefrictional foundation than the assumed cohesive strength of the embankment in this

region. However, this fact was not apparent initially and would not have been uncoveredusing limit equilibrium methods unless anticipated and rigorously investigated. Thesystem that appeared to have a safety factor of 1.12, was in fact at a state of imminentfailure. If the failure had occurred and was back analyzed by the conventionalapproach, not appreciating the subtleties of the true rupture surface, the strength wouldbe found to be 63 kPa (1300 psf), or 81 percent of the actual strength. In this case, thedesign assumption was unconservative because the critical slip surface was not located.The unconservative design assumption leads to a conservative interpretation of strengthin back analysis. Importantly, a limit equilibrium analysis of the surface predicted by

Table 1. Back-calculated strength.

Embankment Strength

Back Calculated

Friction Angle(degrees)

Lower Bound 22-24

Upper Bound (High Friction Angle) 16Upper Bound (High Cohesion) 11

Average 18



numerical modeling leads to a safety factor of 1.0, provided the specific surface isevaluated or a non–circular search routine is adequately constrained to identify thesurface.

Limit EquilibriumSF = 1.12

Zone of High Shear

Strains Indicating

Failure

Spoil from Alluvial Silts

& Clays c=77 kPa

Alluvial Silty Sands phi = 32 deg

Figure 3. Comparison of Failure Surface by Numerical Modeling and Limit Equilibrium Circular Search.

Pore Pressures. Because pore pressures control effective stresses, they impact the interpreted

back-calculated strength. A cross-section of Dam No. 10 on the Kentucky River is shown in

Figure 4a. At the initiation of the remediation for this project there was no subsurface

information for the site, and no original design information. It was known that the structure was

founded on limestone, and that uplift pressures were not considered for design of dams foundedon rock in this era (circa 1905). Previous technical reviews of stability assigned friction angles in

the neighborhood of 45 degrees for the bedrock-concrete interface strength. A detailed

subsurface investigation for the structure showed the limestone below the dam to be highly

fractured. Moreover, extensive thin layers of shale, that had weathered to a clay consistency,

were present at depths of one to two feet below the base of the dam. Laboratory test results

showed that the effective stress friction angle of this clay was on the order of 27 to 30 degrees.

A conventional assessment of the stability of the dam indicates that a friction angle of 43 degrees

is required to maintain stability for the maximum design flood, an event that occurs every year on

average because of the headwater tailwater relationships in this riverine environment. Based on

our interpretation of the foundation strength, we would have calculated a safety factor less than

1.0 for a common loading condition. Given the inconsistency between the required shearstrength and our interpretation of subsurface conditions, we investigated potential limitations in

the analyses. The first thing considered was the assumed pore pressure distribution. The

complete length of dam is a spillway section and difficult access made the installation of

piezometers too costly during the exploration program. Accordingly, the conventional

assumption of a triangular distribution of head loss from heal to toe, with no head loss in the

upstream sediments, was initially assumed, as it was in all previous analyses. These assumptions

are shown in Figure 4b. However, given the fractured nature of the bedrock and the significant

depth of sediment upstream, it is plausible that the majority of head loss is occurring in the



sediment (Reduced pressures in Fig. 4). If this is the case, the required interface strength drops

from 43 to 27 degrees, a substantial range for the variation in possible assumptions of water

pressure. Although the possibility of reduced pore pressure may help explain why the dam could

be stable given the weak clay seams, the three dimensional effects have a much greater influence,

which is discussed in the next section.

a) b)

10.5 m

9.75 m

Interbedded Limestone and Shale

SedimentConcrete

Gravity Dam

Reduced Pressurewith Headlossin Sediment

Sediment

Figure 4. Cross-Section through Kentucky River Dam No. 10.

a) Idealized Section, b) Model for Stability Assessment.

Three-Dimensional or "End Effects."

Stability analyses are conventionally performed on idealized two-dimensional cross-sections,

which are based on plane strain conditions. At Lock and Dam 10, coring through the dam

indicated that the construction joints between concrete monoliths are essentially rubble, and could

not be relied upon as shear connections between monoliths. Although it was considered

imprudent to rely on the shear resistance between monoliths as a design consideration, it wasrecognized that some resistance was likely to be present. An attempt was made to estimate the

magnitude of this resistance in order to understand the inconsistency between required strength

and interpreted strength. Accordingly, numerical modeling with the program FLAC (ITASCA

2002) was used to model the entire dam as a beam. The dam is a spillway over its complete

length of 240 feet; it has a height of 34 feet; a width of 32 feet; and is made up of ten monoliths

24 feet long. The assumption was made that only a nominal frictional resistance (35 degrees) was

available between monoliths (no tensile or cohesive strength) and that the ends were fixed at the

ends. The modeling effort produced a surprising result in which a zone within the dam formed a

compressive arch that developed significant flexural resistance. Figure 5 illustrates the idealized

model of the dam taken in plan view. The dam is attached to an abutment and training wall on

the left, and the lock river wall on the right, both assumed to be stable. The distributed load on

the beam was progressively increased to represent increasing the net hydrostatic pressures fromhigher pools. Based on this analysis, the compressive arch that develops has sufficient capacity

to carry the complete hydraulic load acting on the dam during the maximum design flood,

independent of any frictional resistance at the base, and with only frictional resistance between

monoliths. This example clearly illustrates how difficult it would be to back-calculate strengths if

there is a significant, but uncertain, three-dimensional influence. Although, the three dimensional

effects are extreme in this case, influences of 5 to 30 percent are believed to be expected Azzouz

et. al, (1981).



2 M P a 2 M

P a4 MPa

6 MPa2 M P a

4 MPa

8 MPa

Contour Interval = 2 MPaLock Wall Abutment Wall

Distributed Water Load

Flow

"Compressive Arch"

Frictional Interface Elements

Separating Concrete BlocksMesh Density

Figure 5. Plan View of Kentucky River Dam No. 10, Modeled as a Beam.

SUMMARY

Both a review of the definition of safety factor and specific examples are used to illustrate the

significant challenge in accurately back-calculating shear strength from failures. The examples

provided herein are intended to illustrate that successful back-calculation requires accurate

information regarding geometry, material properties, and pore pressure distribution. In addition,

adjustments must be made to account for the inherent limitations and assumptions of the models

used in the analyses. All of these factors contribute to the results from back-analysis being at best

approximate, and at worst in gross error. Finally, it is important to remember that all assumptions

that are conservative in design are unconservative in back analysis.

REFERENCES

Azzouz, A.S., Baligh, M.M., and Ladd, C.C. (1981). "Three-Dimensional Stability Analyses of Four

Embankment Failures," X Int. Conf. Soil Mech. Found. Engrg, Vol 3, pp 343-346, Stockholm.

Deschamps, R., Hynes, C., and Wigh, R. (1999). "Analysis and Stabilization of a Failing Earth Dam

Founded on Claystone," ASCE, J. Performance of Constructed Facilities, Vol. 13, No. 4, pp. 143-151.

Duncan and Stark, (1992). "Soil Strengths from Back-Analysis of Slope Failures," Proc. Stability and

Performance of Slopes and Embankments II, ASCE, GSP 31, pp. 890-904, Berkeley.

Gilbert, R.B., Wright, S.G., and Liedtke, E. (1998). "Uncertainty in Back Analysis of Slopes: KettlemanHills Case history," J Geotech. Geoenviron. Engrg, ASCE, 124(12), pp1167-1176.

ITASCA (2002) "FLAC Users Manual," Versions 3.4 and 4.0.

Leonards, G.A. (1982). "Investigation of Failures," J. Geotech. Engrg, ASCE, 108(2), 187-246.

Leroueil, S. and Tavenas, F. (1981). "Pitfalls of Back-Analysis," X Int. Conf. Of Soil Mech. and

Foundation Enginnering, Vol 1, pp 185-190, Stockholm.



Stark, T.D. and Eid, H.T. (1998). "Performance of Three-Dimensional Slope Stability Methods in

Practice," J. Geotech. Geoneviron. Engrg., ASCE, 124(11), pp. 1049-1060.

Tang, W.H., Stark, T.D., and Angulo, M. (1998). "Reliability in Back Analysis of Slope Failures," Soils

and Foundations, 39(5), pp73-80.

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Limitations in the Backanalysis of Shear Strength From Failures